A set of Grand Unified Theories based upon the gauge groups SU(5)L × SU(5)R, SO(10)L×SO(10)R and SU(4)C×SU(4)L×SU(4)R is explored. Several novel features distinguish these theories from the well-known SU(5), SO(10) and SU(4)C×SU(2)L×SU(2)R models which they generalize. Firstly, Standard Model quarks and leptons are accompanied by and mix with heavy SU(2)L × SU(2)R singlet partners. The resultin...

Let R be a real closed field. The Pierce-Birkhoff conjecture says that any piecewise polynomial function f on Rn can be obtained from the polynomial ring R[x1, . . . , xn] by iterating the operations of maximum and minimum. The purpose of this paper is threefold. First, we state a new conjecture, called the Connectedness conjecture, which asserts, for every pair of points α, β ∈ Sper R[x1, . . ...

The 2-Interval Pattern problem is to find the largest constrained pattern in a set of 2-intervals. The constrained pattern is a subset of the given 2-intervals such that any pair of them are R-comparable, where model R ⊆ {<, @, () }. The problem stems from the study of general representation of RNA secondary structures. In this paper, we give three improved algorithms for different models. Firs...

The proliferation of location-based service applications has led to a substantial surge in the amount life trajectory data produced by mobile devices. And these frequently contain confidential personal details. Simultaneously, corresponding relatively lagging privacy protection technology and improper handling method will make tremendous problems with breach. Therefore, this paper presents publ...

Let L = − 1 w div ( A ∇ u ) + μ be the generalized degenerate Schrödinger operator in 2 R d with ≥ 3 suitable weight and measure μ. The main aim of this paper is threefold. Firstly, we obtain an upper bound for fundamental solution L. Secondly, prove some estimates heat kernel including bound, Hölder continuity a comparison estimate. Finally, apply results to study maximal function characteriza...

Multiple-Input Multiple-Output (MIMO) radar provides much more flexibility than the traditional radar thanks to its ability to realize far more observation channels than the actual number of transmit and receive (T/R) elements. In designing the MIMO imaging radar arrays, the commonly used virtual array theory generally assumes that all elements are on the same line. However, due to the physical...

For r = (r1, . . . , rd) ∈ R d the map τr : Z → Z given by τr(a1, . . . , ad) = (a2, . . . , ad,−⌊r1a1 + · · ·+ rdad⌋) is called a shift radix system if for each a ∈ Z there exists an integer k > 0 with τ r (a) = 0. As shown in the first two parts of this series of papers shift radix systems are intimately related to certain well-known notions of number systems like β-expansions and canonical n...

where x ∈ R denotes a sample and xj ∈ R denotes the j-th explanatory variable for each j ∈ [1, p] , {j ∈ N|1 ≤ j ≤ p}. This was first proposed by Hastie and Tibshirani (1987) and is known as a generalized additive model (GAM). In this paper, we call fj(·) a weight function and f(·) a GAM predictor. This not only includes linear predictors but also captures nonlinear relationships between explan...

We give complete geometric invariants of cobordisms framed fold maps. These consist two types. take the immersion singular set into target manifold together with information about non-triviality normal bundle in source manifold. were introduced author's earlier works. Secondly we induced stable partial framing on whose studied general by Koschorke. show that these describe completely cobordism ...